Global Dynamics Analysis of Non-Local Delayed Reaction-Diffusion Avian Influenza Model with Vaccination and Multiple Transmission Routes in the Spatial Heterogeneous Environment

Abstract

In order to reveal the transmission dynamics of Avian influenza and explore effective control measures, we develop a non-local delayed reaction-diffusion model of Avian influenza with vaccination and multiple transmission routes in the heterogeneous spatial environment, taking into account the incubation period of Avian influenza in humans and poultry. Firstly, the well-posedness of model is obtained which includes the existence, uniform boundedness and the existence of global attractor. Further, the basic reproduction number \({\mathcal {R}}_0 \) of this model is calculated by the definition of the spectral radius of the next generation operator, and its variational form is also derived. Further, the global dynamics of the model is established based on the biological significance of \({\mathcal {R}}_0 \). To be more precise, if \({\mathcal {R}}_0<1\), the disease-free steady state is globally asymptotically stable (i.e., the disease is extinct), while if \({\mathcal {R}}_0>1\), the disease is uniformly persistent and model admits at least one endemic steady state. In addition, by constructing suitable Lyapunov functionals, we achieve the global asymptotic stability of the disease-free and endemic steady states of this model in spatially homogeneous. Finally, some numerical simulations illustrate the main theoretical results, and discuss the sensitivity of \({\mathcal {R}}_0 \) on the model parameters and the influences of non-local delayed and diffusion rates on the transmission of Avian influenza. The theoretical results and numerical simulations show that prolonging the incubation period, controlling the movement of infected poultry, and regular disinfecting the environment are all effective ways to prevent Avian influenza outbreaks.